
there are class many Woodin cardinals;

\({\mathbf {MAX}(\mathsf {\mathcal {A}})}\)  holds for some \(\in \) -formula \(\phi _\mathcal {A}(x)\) ;

\(\mathbf {NS}_{\omega _1}\)  is saturatedSee [1]} for a discussion of saturated ideals on \(\omega _1\) .;

there exists a filter \(G\)  on \(\mathbb {P}_{\mathrm {max}}\)  meeting all the dense subsets of \(\mathbb {P}_{\mathrm {max}}\)  definable in
\(L(\mathcal {A})\)  (where \(\mathcal {A}\in V\)  is the extension of the formula \(\phi _\mathcal {A}(x)\) ).

